What is the solution for y^{''}-y^'-2y=-2x^3+x^2+4x+1 ?

The solution for y^{''}-y^'-2y=-2x^3+x^2+4x+1 is y=c_{1}e^{2x}+c_{2}e^{-x}+x^3-2x^2+3x-4

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y^{''}-y^'-2y=-2x^3+x^2+4x+1

y^{^{''}} - y^{^{'}} - 2 y = - 2 x^{3} + x^{2} + 4 x + 1

y = c_{1} e^{2 x} + c_{2} e^{- x} + x^{3} - 2 x^{2} + 3 x - 4

Solution steps

y^{''} (x) - y^{'} (x) - 2 y = - 2 x^{3} + x^{2} + 4 x + 1

Solve linear ODE:

y = c_{1} e^{2 x} + c_{2} e^{- x} + x^{3} - 2 x^{2} + 3 x - 4

y = c_{1} e^{2 x} + c_{2} e^{- x} + x^{3} - 2 x^{2} + 3 x - 4

y^{^{''}} - 5 y^{^{'}} + 4 y = x

y^{^{''}} - 4 y^{^{'}} + 3 y = x e^{2 x}

y^{^{''}} + 4 y = 3 sin (x) + 5 cos (3 x)

y^{^{''}} + 4 y = x sin^{2} (x)

y^{^{''}} - 2 y^{^{'}} - 3 y = 4, y (0) = 1, y^{^{'}} (0) = - 1

What is the solution for y^{''}-y^'-2y=-2x^3+x^2+4x+1 ?
The solution for y^{''}-y^'-2y=-2x^3+x^2+4x+1 is y=c_{1}e^{2x}+c_{2}e^{-x}+x^3-2x^2+3x-4